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Commercial banks, investment banks, insurance companies, nonfinancial
firms, and pension funds hold portfolios of assets that may include stocks,
bonds, currencies, and derivatives. Each institution needs to quantify
the amount of risk its portfolio may incur in the course of a day, week,
month, or year.

For example, a bank needs to assess its potential losses in order to
set aside enough capital to cover them. Similarly, a company needs to track
the value of its assets and any cash flows resulting from losses in its
portfolio. An investment fund may want to understand potential losses on
its portfolio, not only to allocate its assets better but also to fulfill
its obligation to make set payments to investors. In addition, credit-rating
and regulatory agencies must be able to assess likely losses on portfolios
as well, since they need to set capital requirements and issue credit ratings.

How can these institutions judge the likelihood and magnitude of potential
losses on their portfolios? A new methodology called value at risk (VAR
or VaR) can be used to estimate these losses. This article describes the
various methods used to calculate VAR, paying special attention to VAR's
weaknesses.

WHAT IS VALUE AT RISK?

Value at risk is an estimate of the largest loss that a portfolio is
likely to suffer during all but truly exceptional periods. More precisely,
the VAR is the maximum loss that an institution can be confident it would
lose a certain fraction of the time over a particular period. Consider
a bank with a portfolio of assets that would like to characterize its potential
losses using VAR. For example, the bank could specify a horizon of one
day and set the frequency of maximum loss to 98 percent. In that case,
a VAR calculation might reveal that the maximum loss is $1 million. Thus,
on average, in 98 trading days out of 100, the loss on the portfolio will
not exceed $1 million over a one-day horizon. But on two trading days in
100, losses will, on average, exceed $1 million.

VAR can be used to assess the potential loss on a portfolio of assets
generally. The user can specify any horizon and frequency of loss that
fits his particular circumstances. But the method of calculating VAR depends
not only on the horizon chosen but also on the kinds of assets in the portfolio.
One method may yield good results with portfolios consisting of stocks,
bonds, and currencies over a short horizon, but the same method may not
work well over longer horizons such as a month or a year. If the portfolio
contains derivatives, methods that differ from those used to analyze portfolios
of stocks, bonds, or currencies may be needed.

VAR FOR A SINGLE SHARE OF STOCK

Ultimately, we want to calculate VAR for a general portfolio of different
assets, such as stocks, bonds, currencies, and options.[1] Let's focus
on the simplest case first: a single stock. A portfolio consisting of one
asset will allow us to consider the different methods for assessing VAR
in a simple context. Then, we can generalize the discussion by considering
how the calculation changes when the institution has a portfolio of many
stocks, bonds, or currencies. Finally, we will consider how the inclusion
of derivatives in the portfolio can dramatically change the methodology
for calculating VAR.

Randomness in the Stock Market. Let's consider a portfolio consisting
of a single share of stock worth $1 at the beginning of trading today.
We want to find the VAR over a one-day horizon at a 98 percent confidence
level, that is, the largest one-day price drop we are likely to see on
98 out of every 100 trading days. Since VAR is essentially a statement
about the likelihood of losses on a stock, we need to characterize the
unpredictability of daily changes in our stock's price.

One way to picture the unpredictability of our stock's return over one
day is to imagine the stock market spinning a roulette wheel. Of course,
this is a fiction, but a useful one: economists have found that stock returns
have a random component.

Suppose there are 100 equally likely outcomes on the wheel, with each
outcome corresponding to a specific percentage daily price change or daily
return for our stock.[2] In general, positive and negative returns are
included on the wheel. To determine the return over one day, the stock
market spins the roulette wheel. If the wheel comes up with a return of
25 percent, our stock would be worth $1.25 at the end of the day. Alternatively,
a spin of the wheel may generate a return of minus 25 percent, in which
case our stock would be worth $0.75 at the end of the day. We can't say
for sure what the daily return will be, but we know that it will be one
of the outcomes on the wheel.

Finding the VAR for our $1 stock is particularly simple if we know the
returns on the roulette wheel. Suppose we look at the outcomes on our roulette
wheel and see that 98 of them involve returns no bigger than minus 30 percent
while two outcomes have returns larger than minus 30 percent. Then we have
found the VAR for our $1 stock: the VAR is $0.30 at a 98 percent confidence
level. We can be confident that 98 days out of 100 our daily stock loss
will be no bigger than $0.30. But two days out of 100, the daily loss may
indeed exceed $0.30.

Summary Measures of Randomness. To find the VAR for our stock,
we needed to know the 100 returns on the wheel. But how do we know what
they are? Imagine that, every day, the market is spinning the wheel behind
a curtain. We can't see the outcomes on the wheel, but we do know which
daily returns were selected in the past--we can look them up in the newspaper.
By categorizing past daily returns, we should be able to infer the outcomes
on the wheel.

For example, if we saw that daily returns of 10 percent occurred on
five trading days in 100, on average, we can assume that five outcomes
on the wheel involve a 10 percent return. Similarly, if changes of minus
5 percent occurred on 10 trading days in 100, on average, a return of minus
5 percent must correspond to 10 outcomes on the wheel. By continuing this
analysis, we can associate price changes with all outcomes on the wheel.
Then we will have reconstructed the wheel that the economy spins daily.
Using our reconstructed wheel, we can easily find the VAR.

A simpler way to do this reconstruction is to summarize the 100 returns
on the wheel by using two numbers: the average return (mean) and the volatility
(variance) of the returns. Elementary statistics teaches that if the returns
follow a certain pattern, called the normal, or bell-shaped, distribution,
all the outcomes on the wheel can be summarized by these two numbers.

We can estimate the average return as an equally weighted average of
past daily returns selected by the roulette wheel, returns that, again,
could be looked up in the newspaper. For technical reasons, analysts often
don't perform this calculation but assume instead that the average return
is zero.[3] The second number, the volatility, tells us how much the return
is likely to deviate from its average value for any particular spin. The
volatility, then, measures the capacity of the roulette wheel to generate
extreme returns, whether positive or negative, with respect to the average
value of zero. The higher the volatility of the roulette wheel, the more
it tends to select large returns. We can estimate the volatility as an
equally weighted average of past squared returns. We could use the same
returns we looked up in the newspaper; we only need to square each change.

Armed with the average return of zero and the volatility of our stock's
returns, we can find the VAR over a one-day horizon at the 98 percent confidence
level by following a simple procedure. To calculate VAR for our stock,
we need only multiply today's stock price of $1 times the square root of
the volatility times a number corresponding to the 98 percent confidence
level, called the confidence factor. The confidence factor is derived from
the properties of the normal distribution. At the 98 percent confidence
level, it equals 2.054.[4]

This procedure can be done on any day in the future as well. Let's assume
that it's now tomorrow and the stock price is $0.95. If we wanted to calculate
VAR, we would follow the same procedure as before but use a stock price
of $0.95. We don't need to change the volatility or the confidence number:
they don't vary from day to day. When VAR is calculated in this fashion,
we are using a constant volatility method.

Time-Varying Volatility. The problem with the constant volatility
method is that substantial empirical evidence shows volatility is not constant
from day to day but rather varies over time.[5] A look at a graph of the
daily dollar return on the deutsche mark shows that volatility tends to
cluster together (Figure 1). Notice that highly volatile times, characterized
by large up-and-down swings in the exchange rate, tend to follow one another,
while quiet periods, characterized by smaller up-and-down swings, tend
to follow each other as well. For example, volatility seems to have been
higher in 1991 than in 1990. A graph of the daily return on the S&P
500 confirms this impression for stock prices (Figure 2). The increase
in volatility is particularly apparent after the stock market crash in
1987. Time-varying volatility seems to be a general feature of asset prices
that is seen not only in currencies but also in stocks. Consequently, using
the constant volatility method to calculate VAR could be very misleading.

What does time-varying volatility mean for our roulette wheel analogy?
When the average return and the volatility don't vary from day to day,
the returns on the wheel don't vary either. Thus, the market is spinning
the same roulette wheel every day. But if the volatility is changing from
day to day (time-varying volatility), the returns on the wheel must also
be changing; therefore the market is spinning a different wheel each day.

If the market spins a different roulette wheel every day, VAR becomes
more complicated. How do we know which returns will be on the wheel today?
Equivalently, how do we know today's volatility? The most common solution
to this problem was introduced in 1986 by economist Tim Bollerslev, who
generalized work done by economist Robert Engle in 1982. Bollerslev's time-varying
volatility technique, called the GARCH method, allows us to base our knowledge
of today's roulette wheel on yesterday's wheel.

Bollerslev's GARCH technique estimates the volatility of today's roulette
wheel using yesterday's estimate of volatility and the squared value of
yesterday's return. If yesterday's return was large, in either a positive
or negative direction, and yesterday's volatility was high, today's roulette
wheel will tend to have a high volatility. Thus, today's spin of the wheel
will tend to produce large returns as well. In this way, large returns,
positive or negative, would tend to follow one another, leading to periods
of high and low volatility as we saw in Figures 1 and 2.

How can we estimate today's volatility and find the VAR using Bollerslev's
GARCH method? The daily volatility using GARCH turns out to be a weighted
average of past squared returns, just as it was in the constant volatility
case. The difference is that the constant volatility method weights past
squared returns equally while Bollerslev's GARCH method weights recent
squared returns more heavily than distant returns.

It is easy to calculate volatility using the constant volatility method.
Bollerslev's GARCH method is much harder to implement: to find the right
weight for each past squared return, we must employ a complicated, computer-intensive
procedure. Once we have found today's volatility, we can multiply the confidence
factor times the square root of today's volatility times today's stock
price to find today's VAR. When we use Bollerslev's GARCH method, the confidence
factor is the only number that does not change daily.

RiskMetricsTM. Bollerslev's GARCH method has
found widespread empirical support among financial economists, but the
difficulty in estimating daily volatilities has slowed its adoption by
many institutions engaged in risk management. To make the calculations
easier, J.P. Morgan introduced RiskMetricsTM, a risk management
system that includes techniques to approximate GARCH volatilities. Like
Bollerslev's method, the RiskMetricsTM estimate of daily volatility
involves a weighted average of past squared returns, with recent squared
returns weighted more heavily. The RiskMetricsTM weights are chosen
to produce daily volatility estimates similar to GARCH volatilities. The
set of weights calculated by the RiskMetricsTM method is easier
to compute and can be used for any asset in the portfolio. For example,
the analyst would use the same set of weights to calculate volatilities
of stocks, bonds, and currencies. Bollerslev's GARCH method, in contrast,
requires the computation of different weights for each volatility calculation,
and each set of weights is harder to calculate than they would be using
the RiskMetricsTM method.[6]

Other Methods. Two other methods of calculating volatility are
sometimes used. The first method relies on recognizing that pricing methods
for options require the user to specify his estimate of the future volatility
of an asset. For example, if a user wants to price an option on a stock
using a method such as the popular Black-Scholes method, he must specify
an estimate of the volatility of the stock over the life of the option.[7]
Since option prices are observable in the marketplace, the market's view
of volatility can be backed out of the option price using the Black-Scholes
formula. Volatility estimates inferred from option prices in this way are
called implied volatilities.

This method has two disadvantages that limit its appeal. First, options
may not be traded on the particular asset of interest. Thus, implied volatility
estimates may not be obtainable for some assets in the portfolio. Second,
economists are unsure about whether implied volatility estimates are better
than GARCH estimates of daily volatility.

The other method of estimating volatility is based on judgment. The
user analyzes the economic environment and forecasts volatility based on
his subjective views. This method has limited appeal as well, since testing
the validity of a subjective view is difficult.

VAR FOR A PORTFOLIO OF ASSETS

Up to this point, we have considered only how to calculate the VAR of
a portfolio consisting of a single stock. Now let's look at a portfolio
of two stocks. The principles we are about to discuss apply generally to
portfolios of many assets, but we will consider just two stocks to make
the ideas clear.

As before, ultimately we want to find the volatility of the return on
the portfolio. It's clear that the volatility of the portfolio should depend
on the volatility of the return of each stock in the portfolio. So, we
need to estimate the volatilities of the returns of both stocks. But stock
returns may covary as well. For example, if the covariance between the
stocks in a portfolio of two stocks is negative, then when one stock has
a positive return, the other has a negative return, and vice versa.

Thus, the two stocks dampen each other's swings in return, producing
a portfolio whose volatility is lower than the volatility of each stock
in the portfolio. Adding more stocks to the portfolio would reduce the
volatility further, provided the additional stocks' returns are not highly
positively correlated with the return of the initial portfolio. To account
for this effect, we must also estimate the covariance between the stocks'
returns. Once we know the stock returns' volatilities and covariances,
we can calculate the volatility of the entire portfolio and find the VAR
as before.

As an example of the calculation, suppose we have invested $1 in stocks
1, 2, and 3. Then by an elementary statistical formula, the daily volatility
of the portfolio would be

volatility(portfolio)=volatility(stock 1) +

volatility(stock 2) + volatility(stock 3) +

2.0 x covariance(stock 1,stock 2) +

2.0 x covariance(stock 1, stock 3) +

2.0 x covariance(stock 2, stock 3)

Notice that if the covariance between the daily returns of stocks 1,
2, and 3 were zero, we could sum the volatilities of each stock to get
the volatility of the portfolio. Thus, if covariances between all assets
were zero, we could find the VAR of each asset separately and then sum
them to get the VAR of the portfolio. But since covariances are, in general,
not zero, we can't, in general, find the VAR of individual assets and sum
them to get the VAR of the portfolio. Moreover, we can't find the VARs
of asset classes such as stock and currency portfolios separately and sum
them. We must account for the covariances between asset classes as well.

To calculate covariances between the assets' returns using the constant
covariance method, we use an equally weighted average of the products of
each stock's past daily returns. However, since economists have found evidence
that covariances change over time, it may be advisable to estimate time-varying
covariances using an extension of Bollerslev's GARCH method or the RiskMetricsTM
GARCH approximation.[8]

WHAT ABOUT DERIVATIVES?

Many portfolios have significant numbers of derivatives such as futures,
options, and swaps, all of which are securities whose value is derived
from the value of some other asset. Consider a derivative on our $1 stock.
We know how to find the VAR of the stock over a one-day horizon at the
98 percent confidence level: we find the volatility of its return and multiply
its square root by the product of today's stock price and the confidence
factor. But how can we find the VAR of a derivative on this stock?

One method is to link the derivative to the underlying stock and use
the standard VAR method. To do this, we use a derivative-pricing method,
such as the Black-Scholes model, to calculate a number called delta, which
gives us a way to translate the derivative portfolio into the stock portfolio.
A derivative's delta tells us how the derivative's price changes when the
stock price changes a small amount. For example, if the delta is 0.5, the
derivative's price goes up half as much as the stock's price. For small
price changes, a derivative with a delta of 0.5 behaves as if it is half
a share of the $1 stock. So, using our estimate of the stock's volatility,
we could calculate VAR as we did before: by multiplying $0.50 times the
square root of the stock's volatility times the confidence factor.

A serious drawback to this method is that it works well only when stock
price changes are small. For larger changes, delta itself can change dramatically,
leading to inaccurate VAR estimates. In general, we need to account for
how delta changes, considerably complicating the analysis.

To avoid this complication, risk managers often use an alternative method
called Monte Carlo analysis. Using the volatility and covariance estimates
for the derivatives' underlying assets as well as a derivative pricing
tool such as the Black-Scholes method, risk managers construct a new roulette
wheel. The new wheel will still have 100 numbers, but each number will
correspond to a potential change in the derivative's price. The computer
can then look at the largest loss the derivative will sustain for 98 of
the outcomes. Let's suppose this loss is $0.01. Then the VAR of the derivative
over a one-day horizon at the 98 percent confidence level is $0.01. Since
RiskMetricsTM yields volatility and covariance estimates, Monte
Carlo evaluation of derivative portfolios can be done under J.P. Morgan's
system as well.[9]

WEAKNESSES OF VAR

When properly used, VAR can give an institution an idea about the maximum
losses it can expect to incur on its portfolio a certain fraction of the
time, making VAR an important risk-management tool. Using VAR calculations,
an institution can judge how it should re-allocate the assets in its portfolio
to achieve the risk level it desires. But VAR methodology is not without
its weaknesses, and, improperly used, it may lead an institution to make
poor risk-management decisions. This can happen for one of two reasons:
either the VAR is incorrectly calculated or the VAR is correctly calculated
but irrelevant to the institution's real risk-management goals.

What Is the Best Method for Estimating Volatility? Bollerslev's
GARCH method works better for currencies than it does for stock prices.
Financial economists have found that stock volatility goes up more as a
result of a large negative return than it does as a result of a large positive
return. A weakness of Bollerslev's GARCH method is that GARCH volatility
estimates don't depend on whether yesterday's return was positive or negative.
Thus, this method can't allow for stock volatility's asymmetric response
to past returns.

To account for this effect, financial economists have developed methods
for estimating asymmetric volatilities.[10] These methods are important
because they can give very different estimates of volatility for days following
large stock returns than would the GARCH or RiskMetricsTM method.
For small daily returns, Bollerslev's method, RiskMetricsTM, and the asymmetric
volatility method yield similar one-day-ahead volatility predictions, leading
a user to think, perhaps, that one model is as good as the others for daily
volatility predictions. But for large daily returns, the one-day-ahead
volatility predictions of these methods can be substantially different.
If an asymmetric volatility method is appropriate for stock prices, both
Bollerslev's method and RiskMetricsTM may understate one-day-ahead
volatility whenever a large drop in stock prices occurred the previous
day, thus producing a potentially substantial underestimate of daily VAR.
Similarly, the GARCH or RiskMetricsTM method could overestimate
the VAR after a large increase in stock prices.

Robert Engle and Victor Ng have provided evidence that a particular
asymmetric volatility method well describes the volatility of Japanese
stock returns and that GARCH methods can substantially underpredict volatility
following large negative returns. Thus, VAR estimates of stock portfolios
produced by GARCH or the RiskMetricsTM GARCH approximation should
be viewed with caution if the calculations are done on days with large
stock returns.

Although having the right method for calculating the volatilities of
assets is important, correctly calculating the covariances between the
returns on assets is also important. Unfortunately, not as much work has
been done by financial economists to identify the right method for calculating
covariances. To date, many methods have been proposed, but no consensus
has yet emerged.

Thus, we don't yet know for sure how we should handle covariances in
portfolios. This uncertainty introduces the risk that any method we use
may substantially under- or overestimate VAR. In particular, RiskMetricsTM
commits the user to a special case of Bollerslev's GARCH method. Since
we don't yet know whether Bollerslev's GARCH method is adequate in describing
covariances, we should use even more caution in interpreting results whenever
we have used covariances in our VAR calculations.

In the long run, the volatility estimates produced by GARCH methods
tend, in general, to approach the values that the constant volatility method
would have calculated. Thus, for horizons much longer than one day, using
the constant volatility method to calculate VAR may be warranted.[11]

Frequency of Large Returns. Using either Bollerslev's GARCH model
or the constant volatility method, we could find the VAR by assuming that
the returns on the wheel follow a normal distribution. However, a substantial
amount of evidence indicates that the normal distribution is inadequate
because large daily returns, positive or negative, occur more often in
the market than a normal distribution would suggest. One remedy is to use
a different distribution for the price changes, one that generates more
frequent large returns.[12] Alternatively, we could use statistical methods
that assume the returns follow the normal distribution, but which remain
valid even if this assumption is mistaken.

Whichever method we use, we are essentially looking at the past frequencies
and magnitudes of returns and attempting to represent them on a reconstructed
wheel. Even if we account for the nonnormality of returns during this process,
there is still a problem: we're going to put on the wheel only those returns
we saw in the past with the frequency we saw in the past. So, if some potential
negative returns are rare or have not yet occurred, we may underrepresent
them on the wheel, implying that the VAR will be underestimated.

Structural Shifts in the Economy. VAR may also be underestimated
if the wheel the market is spinning suddenly changes in an unpredictable
way because of a structural change in the underlying economy. For example,
consider the European Exchange Rate Mechanism (ERM), which kept daily returns
of major European currencies small. In 1993, in response to economic pressures,
much larger returns were suddenly allowed. Thus, the volatility of the
returns suddenly shot up faster than Bollerslev's GARCH method would have
forecast based on past volatilities and returns. If we had calculated the
VAR the day before the shift, we would have underestimated it because we
would have used an estimate of the volatility that was too low. More subtly,
since we never know when the economy may suddenly shift to higher or lower
volatility as a result of a structural change, we will incorrectly estimate
the VAR unless we explicitly account for this possibility.

Because of the problems caused by infrequent large returns and structural
shifts in the economy, it seems prudent, then, to supplement statistical
calculations of VAR with judgmental estimates. For example, an institution
could have asked its economists to project the likely price effects if
the ERM suddenly allowed larger price changes. These projections could
be based on similar historical episodes, economic theory, and empirical
experience. VAR estimates based on judgment could be generated for changes
in central bank monetary regimes, political instability, structural economic
changes, and other events that have either never happened or happen infrequently.

Liquidity of Assets. VAR measures the maximum loss that an institution
can expect a certain fraction of the time over a specific horizon. Losses
are measured by assuming that the assets can be sold at current market
prices. However, if a firm has highly illiquid assets--meaning that they
cannot quickly be resold--VAR may underestimate the true losses, since
the assets may have to be sold at a discount.

Credit Risk. Another potential problem for VAR is that the methods
used to evaluate the assets in the portfolio may not properly treat credit
risk. Suppose a bank buys a portfolio of derivatives from many different
firms. The derivatives are valuable to the bank because they impose obligations
on the firms. For example, one of the derivatives may obligate a firm to
sell foreign currency to the bank at a price below the current market price,
yielding a profit to the bank under some conditions, but it may also obligate
the bank to deliver foreign exchange at a below-market price under other
conditions. Using the Black-Scholes method and a Monte-Carlo simulation,
which assume no derivative credit risk, the bank calculates a VAR of $5
million at a 98 percent confidence rate for a three-month horizon. But
if some of the firms may default on their obligations, the true value of
these derivatives is lower than would be estimated by the Black-Scholes
method coupled with Monte-Carlo analysis. Thus, the true value at risk
is larger than $5 million. To account for this possibility when valuing
derivatives, the bank should use a method that includes credit risk. For
some applications, credit risk may be small enough to ignore, but, in general,
users need to include credit risk analysis in their VAR methods.

Is VAR the Right Methodology? In many situations, VAR may not
be the correct risk-management methodology. If we pick a specific loss
such as $1 million, VAR allows us to estimate how often we can expect to
experience this particular loss. For example, using VAR we might estimate
that we will lose at least $1 million on one trading day in 20, on average.
During some 20-day periods, we might lose less than $1 million. During
other 20-day periods, we might lose more than $1 million on more than one
day. VAR tells us how often we can expect to experience particular losses.
It doesn't tell us how large those losses are likely to be. In particular,
in any 20-day period, there is always one day on which the worst loss is
experienced. If we want to know the size and frequency of the worst loss,
VAR provides no guidance.

One way of handling this is to use worst-case-scenario analysis (WCSA),
proposed by Jacob Boudoukh, Matthew Richardson, and Robert Whitlelaw. WCSA
might show that on the day with the worst price change in a 20-day period,
we can expect to lose at least $2.77 million 5 percent of the time, a number
substantially bigger than $1 million. Thus, if a firm is interested in
the size of a worst-case loss, VAR could underestimate it.

CONCLUSION

VAR is an important new concept in portfolio risk management. It gives
the maximum loss that an institution can expect to lose with a certain
frequency over a specific horizon, and it can be calculated by using a
constant volatility or time-varying volatility method. There are, however,
problems in implementation and interpretation. To implement VAR calculations,
it is important to use the right method, especially under unusual circumstances
such as stock market crashes. Although much progress has been made in describing
how volatilities change through time, not as much progress has been made
in the description of time-varying covariances. Thus, VAR numbers should
be viewed with caution at this point.

Besides the problem of identifying the right method, VAR measures may
mislead unless they properly account for liquidity risk, rare or unique
events, and credit risk. In many situations, it may not be the right risk-management
concept. An institution may want to investigate an alternative, such as
worst-case-scenario analysis.

Despite the contribution that VAR can make to a firm's understanding
of the risks in its portfolio, these risks can be misunderstood if they
are not communicated effectively to a management that understands the value
and limitations of sophisticated financial technology. Poor management
practices, which could lead to unauthorized trades, may also contribute
to this misunderstanding. Thus, a firm should use VAR in the context of
a broader risk-management culture, fostered not only by the firm's risk
managers but also by its senior management.

FOOTNOTES

*Greg Hopper is an economist in the Research Department of the Philadelphia
Fed.

1 An option is a derivative security, i.e., its value is derived from
the value of some other asset.

2 In reality, when economists imagine stock returns on a wheel, they
think of the wheel as having an infinite number of outcomes so that all
possible returns are represented. To simplify the discussion, I have used
100 outcomes on the wheel as an approximation to an infinite-outcome wheel.

3 Since the average return is estimated very imprecisely, it may pay
to set it to zero to avoid corrupting the rest of the VAR analysis. For
more discussion on setting the average return equal to zero, see the articles
by Steven Figlewski and David Hsieh (1995).

4 From elementary statistics, 2.054 standard deviations leave 2 percent
of the normal distribution in its left tail, which corresponds to stock
losses occurring 2 percent of the time. If the confidence level were 95
percent, the confidence factor would be 1.65, because 1.65 standard deviations
leave 5 percent of the normal distribution in the left tail.

5 The evidence suggests that volatility is time-varying for short horizons
such as up to a week or 10 days. For longer horizons, the evidence for
time-varying volatility is weaker. If a firm is interested in calculating
VAR over a much longer horizon, the time-varying volatility issue may not
be so important.

6 Under the RiskMetricsTM method, a different set of weights
is calculated for each of a series of over 400 assets. The weights are
then combined to yield a single composite set of weights that can be used
for any asset in the portfolio.

7 For an explanation of this method, see the article by Fischer Black
and Myron Scholes.

8 For further discussion on covariance GARCH techniques, see the paper
by Robert Engle and Kenneth Kroner and the 1990 paper by Tim Bollerslev.

9 For more detail on this process, see the RiskMetricsTM technical document.
For an example of a related methodology, see the 1993 articles by David
Hsieh.

10 The prototypical asymmetric volatility model is EGARCH. See the article
by Daniel Nelson.

11 See the article by David Hsieh (1993a) for a discussion about when
the constant volatility model may be appropriate.

12 For an example of this technique, see the article by Daniel Nelson.

REFERENCES

Academic Literature:

Black, Fischer, and Myron Scholes. "The Pricing of Options and Corporate
Liabilities," Journal of Political Economy, 81 (1973), pp. 637-59.